Related papers: Polynomial method in coding and information theory
Efficient and stable algorithms for the calculation of spectral quantities and correlation functions are some of the key tools in computational condensed matter physics. In this article we review basic properties and recent developments of…
This paper investigates universal polar coding schemes. In particular, a notion of ordering (called convolutional path) is introduced between probability distributions to determine when a polar compression (or communication) scheme designed…
We establish sharp estimates that adapt the polynomial method to arbitrary varieties. These include a partitioning theorem, estimates on polynomials vanishing on fixed sets and bounds for the number of connected components of real algebraic…
Let $A(n,d,w)$ be the largest possible size of an $(n,d,w)$ constant-weight binary code. By adding new constraints to Delsarte linear programming, we obtain twenty three new upper bounds on $A(n,d,w)$ for $n \leq 28$. The used techniques…
The complexity of computing the solutions of a system of multivariate polynomial equations by means of Groebner bases computations is upper bounded by a function of the solving degree. In this paper, we discuss how to rigorously estimate…
In the paper, we introduce a matrix method to constructively determine spaces of polynomial solutions (in general, multiplied by exponentials) to a system of constant coefficient linear PDE's with polynomial (multiplied by exponentials)…
This paper contains the results collected so far on polynomial composites in terms of many basic algebraic properties. Since it is a polynomial structure, results for monoid domains come in here and there. The second part of the paper…
Motivated by the problem of the existence of bounds on degrees and orders in checking primality of radical (partial) differential ideals, the nonstandard methods of van den Dries and Schmidt ["Bounds in the theory of polynomial rings over…
Multivariate polynomial optimization is a prevalent model for a number of engineering problems. From a mathematical viewpoint, polynomial optimization is challenging because it is non-convex. The Lasserre's theory, based on semidefinite…
The Poltyrev bound provides a very tight upper bound on the decoding error probability when using binary linear codes for transmission over the binary symmetric channel and the additive white Gaussian noise channel, making use of the code's…
Numerical homotopy continuation methods for three classes of polynomial systems are presented. For a generic instance of the class, every path leads to a solution and the homotopy is optimal. The counting of the roots mirrors the resolution…
We obtain a new coding and decoding method using the generalized Pell $(p,i)$ -numbers. The relations among the code matrix elements, error detection and correction have been established for this coding theory. We give two new blocking…
This work introduces a decoding strategy for binary self-dual codes possessing an automorphism of a specific type. The proposed algorithm is a hard decision iterative decoding scheme. The enclosed experiments show that the new decoding…
A general method of coding over expansion is proposed,which allows one to reduce the highly non-trivial problems of coding over analog channels and compressing analog sources to a set of much simpler subproblems, coding over discrete…
Capacity formulas and random-coding exponents are derived for a generalized family of Gel'fand-Pinsker coding problems. These exponents yield asymptotic upper bounds on the achievable log probability of error. In our model, information is…
This paper has two contributions. First, we propose a novel coded matrix multiplication technique called Generalized PolyDot codes that advances on existing methods for coded matrix multiplication under storage and communication…
Solving polynomial equations is a subtask of polynomial optimization. This article introduces systems of such equations and the main approaches for solving them. We discuss critical point equations, algebraic varieties, and solution counts.…
We extend the Ax-Katz theorem for a single polynomial from finite fields to the rings Z_m with m composite. This extension not only yields the analogous result, but gives significantly higher divisibility bounds. We conjecture what computer…
We consider three classification systems for distributed decision tasks: With unbounded computation and certificates, defined by Balliu, D'Angelo, Fraigniaud, and Olivetti [JCSS'18], and with (two flavors of) polynomially bounded local…
Arithmetic combinatorics is often concerned with the problem of bounding the behaviour of arbitrary finite sets in a group or ring with respect to arithmetic operations such as addition or multiplication. Similarly, combinatorial geometry…